Process equations
Stochastic cohort model with a plus group to model the unobserved states: \[\log(N_{a,y}) = \left\{\begin{matrix}
\log(N_{a-1,y-1}) - Z_{a-1,y-1} + \delta_{a,y}, & a < A \\
\log\{N_{a-1,y-1}\exp(-Z_{a-1,y-1}) + N_{a,y-1}\exp(-Z_{a,y-1})\} + \delta_{a,y}, & a = A.
\end{matrix}\right. \]
The ages are 1-10+ and years are 1975-2015. \(Z_{a,y} = F_{a,y} + M_{a,y}\), where \(M_{a,y} = 0.2\) is the base case assumption.
Recruitments, \(N_{1,1}, ... N_{1,Y}\), are treated as uncorrelated lognormal random variables \[\log(N_{1,y}) \overset{iid}\sim N(r, \sigma_{r}^2).\]
Catches are modeled using the Baranov catch equation, \[C_{a,y} = N_{a,y}\{1 - \exp(-Z_{a,y})\}F_{a,y}/Z_{a,y}.\]
Fishing moralities are modeled as a stochastic process, with \[Cov\{\log(F_{a,y}),\log(F_{a-j,y-k})\} = \frac{\sigma_{F}^2 \varphi_{F,a}^j \varphi_{F,y}^k}{(1-\varphi_{F,a}^2)(1-\varphi_{F,y}^2)}.\]
Observation equations
The model predicted catch for survey \(s\) is \[\log(I_{s,a,y}) = \log(q_{s,a}) + \log(N_{a,y}) - t_{s,y}Z_{a,y} + \varepsilon_{s,a,y}, ~~ \varepsilon_{s,a,y} \overset{iid}\sim N(0, \sigma_{s,G(a)}^2).\]
Survey variance was split out and self-weighted by age groups 1-3, 4-7, and 8-10+
Total catch and age compositions were treated separately. Total catch was modeled as lognormal, \[\log(C_{obs,y}) = \log(C_{y}) + \varepsilon_{C,y}, ~~ \varepsilon_{C,y} \overset{iid}\sim N(0, \sigma_{C}^2)\] Age compositions were modeled as multiplicative logistic normal with a censored component for zero’s